SUKFACES IN HYPERSPACE. 291 



Xij = dXj/dxj, would form a system of the second order. Let us 

 consider the transformation of this system. 



oxj dXj dxj axj 



ay I dXj dXj 



= St^-T;. (X,0 + 2, (Z,) -^ . (31) 



If it were not for the second term, the transformation would be co- 

 variant, but the presence of this term shows that the derivatives of a 

 covariant system of the first order do not form a covariant system of 

 the second order. 



The same is true for covariant systems of any order, — their deri- 

 vatives do not form a covariant system. For instance in the case of a 

 covariant system of the second order Z„, by a similar transformation. 



d^Vi %;•_!_ d^y, dl/i' 



dXrs _ ^ d(Xij) _i_ ^ /-rr \ 



•^; ^HkyriYsiytk — T — -r Zij\Aij) 



oxt dyk 



where dyi/dxs = ysi and d^yi/dxrdxt = djri/dxt. 



_dXrdxt dxs dXgdxt dxr. 



, (31') 



The fundamental relation dxi = l^idjdyj may be written in matrical 

 notation as dx = c?yVj,x. It follows that Mc = V^x. We may also 

 write dy = dx-VxY- Hence VxY and VyX are reciprocals. The rela- 

 tions (29) and (30) may be written as 



X = V.y(X), XY = VxyV.y:(XY), 

 (X) = V,,x-X, (XY) = V^xV^x:XY, 

 X° = (X°).V,x, X°Y° = (X°Y°):V,xV„x, 

 (X°) = X°.v.y, (X°Y°) = X°Y°:VxyV.y, 



and so for systems of any order. 



The differentiation of a system of the zeroth order/ is accomplished 

 as: 



df = d(f), dx'Vj = dyVM 



dx'Vj = dx'V^yVyif), V/ = V.y(V/). 



